Internat. J. Math. & Math. Sci.
(1987) 315-320
315
Vol. i0 No. 2
A MONOTONE PATH IN AN EDGE-ORDERED GRAPH
A. BIALOSTOCKI
Department of Mathematics and Applied Statistics
University of Idaho
Moscow, Idaho 83843
and
Y. RODITTY
School of Mathematical Sciences
Tel-Avlv University
Tel-Aviv, Israel 69978
(Received January 8, 1986 and in revised form September 23, 1986)
ABSTRACT.
f
An edge-ordered graph is an ordered pair (G,f), where
in
(G,f)
is a simple path
f({vl,vi+l})
Pk+l:VlV2...Vk+l
f({vi+l,Vi+2})
<
IE(G) I}.
{1,2
is a bljective function, f:E(G)
It is proved that a graph
>
is a graph and
k
such that either
f({vi+l,Vl})
for
i
1,2
k-l.
has the property that (G,f) contains a monotone
G
path of length three for every
G
in
f({vi,vi+l})
or
G
A monotone path of length
f
G
iff
contains as a subgraph, an odd cycle of
length at least five or one of six listed graphs.
KEY WORDS AND PHRASES.
Edge-orded graph, monotone path.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
05C55, 05C58.
INTRODUCTION.
Graphs in this paper are finite, loopless and have no multiple edges.
by
G
G(V,E)
as its vertex-set.
on
n
E(G)
a graph with
Kn’ Pn
Let
vertices, respectively.
x(G), and
d(v)
a subgraph of
C
as its edge-set of cardlnallty
and
H
G
We denote
and
V(G)
be the complete graph, the path and the cycle,
n
The vertex-chromatlc number of
is the degree of a vertex
G
e(G)
V(G).
v
By
H
G
G
is denoted by
we mean that
H
is
is the negation of this fact.
Definitions and Notation
I.
An edge-ordered graph is an ordered pair
is a bijective function, f:E(G)
2.
path
{1,2,3
A monotone path of length k, k
Pk+l:VlV2...Vk+l
G
in
f({vl,vi+l})
<
(G,f), where
G
is a graph and
f
e(G)}.
3
in
such that either
f({vi+l,Vi+2})
(G,f), denoted by
MPk+ I,
is a simple
A. BIALOSTOCKI and Y. RODITTY
316
or
f({vi,vi+l}
3.
function
MP
G
We denote by
f({vi+l,Vi+2}
>
for
the fact that
k
1,2
i
(G,f)
k-l.
MP
contains an
k
for every
f, and let
{GIG
MPk},
3
k
The following Theorem i.I is well known, see [I], [2], [3], for a proof and
generalizations:
THEOREM I.I.
such that
For every positive integer k, there is a minimal integer
Kn Ak
for every
g(k),
a g(k).
n
The main result of this paper is:
A graph
THEOREM 1.2.
G
belongs to
A
4
iff
G
contains either
C2n+l,
n
2,
or one of the following graphs:
Fig.
REMARK.
2.
G
Notice that a graph
belongs to
A3
iff
G
contains a path
P3"
PROOFS
The following lemmas are essential for the proof of Theorem 1.2.
LEMMA 2.1.
to
A
H
The graphs
H 2, H 3, H 4, H 5, H 6,
and
C2n+l
where
n Z 2
belong
4.
PROOF.
prove that
there is an
The proof is a straightforward verification for each of the graphs.
H
4
f
A
4.
isomorphism, the integers
following 5 ways:
Assume that
The proof of the remaining cases is similar.
such that no
MP
1,2,3
4
occurs in
(H4,f).
It turns out that up to
can be assigned to the edges of
H
4
in the
We
A MONOTONE PATH IN AN EDGE-ORDERED GRAPH
317
Now, one can see that in each case it is impossible to complete the labeling of the
(H4,f)
edges such that
MP
does not contain an
4.
The following definition is needed for the next lemma.
DEFINITION.
m
0
and
R2n(al,a 2
n
Let
2.
a2n)
a,b,Cl,C2,...,Cm+l,a
,a2n be non-negative integers where
The graph Ll(m,a,b,cl,c
Cm+l), L2(a,b), L3(a,b), and
2
are defined in Fig. 2.
edges
edges
Lz(.b)
L$(a .b)
eZn
L1(m,a,b,cl,c
edges
,Cm
edges
ItZn(a! ’aZ
’ab,)
edges
A. BIALOSTOCKI and Y. RODITTY
318
LEMMA 2.2.
where
m
0
and
do not belong to
(ii).
PROOF.
For all non-negative integers
(i).
2, the graphs
n
A
4
e(G).
for
e
K
A
4.
For the proof of (1), a partial labeling of the
does not belong to
edges of the graphs in question is presented in Fig. 3.
edges is arbitrary.
a2n
a2n
4.
The complete graph
We set
L I,
a,b,Cl,C2,...Cm+l,a
L2(a,b), L3(a,b), and R2n(al,a 2
An
MP
4
will not occur.
in Fig. 3.
Fi.
3a
The labeling of the remaining
A labeling of
E(K 4)
is also presented
A MONOTONE PATH IN AN EDGE-ORDERED GRAPH
x
Y
-
m even
c
r
Z-
"
319
2
Wt odd
"
w even
3
ta
Odd
2
W even
_l+
e_
tm
Odd
LI L1(m’a’b’c1’c2
Cm*1
Fig. 3b
PROOF OF THEOREM 1.2.
an
H
i
1,...,6
i
We may assume that
x(G)
two cases:
CASE I.
path.
If
Let
t
Clearly, every graph
belongs to
A
4.
G
is connected and contains a
2
and
X(G)
4, then
2.
G
x(G)
If
is a tree, let
is double star yielding
Pt
5.
Note that the maximality of
to
Pt’
say
Pn’
3
i
t-2
there is a vertex-disjoint path to
where
n
x i, then
C2n+l,
n
P4’
hence
x(G)
2.
2, or
G e A
4.
We consider
Pt:XlX2
x
t
be its longest
A4, a contradiction.
G
Hence,
implies that there is no vertex-disjolnt path
3, with initial vertex
HI
that contains
3.
G
t
initial vertex is
G
To prove the opposite containment let
Pt’
x
2
say
G, and we are through.
or
xt_ I.
Pm’
where
If for a certain i,
m
Otherwise, G
3, whose
can be embedded
A. BIALOSTOCKI and Y. RODITTY
320
R2n(al,a 2
in a graph
G
that
Let
be the shortest cycle in
One can see that if
some non-negative integers
H
H
it follows that if
integers
Case
al,a 2
,a2t
G
G.
G
2
and
A 4, a contradiction.
Thus we may assume
3.
t
G
then
Assume first
t
2, i.e.,
H
G
R4(al,a2,a3,a 4)
3
G
then
and hence by Lemma 2.2, G
al,a2,a3,a 4
Thus we may assume that
tion.
and non-negatlve integers
n
is not a tree.
C2t
4-cycle.
for a certain
and in view of Lema 2.2, G
a2n
al,a 2,
a2n)
is a
for
A4, a contradic-
Similarly in view of the mlnlmallty of
R2t(al,a2,...,a2t
implying that
C2t
G
C2t
for some non-negatlve
A4, a contradiction.
Hence, the proof of
is completed.
Let
CASE 2.
we are through.
G
triangle in
Let
(i
through, or
H
that
Hence
6
(G)
3.
Hence
with a vertex-set
d(x), d(y), d(z)
G
K4
G.
L2(O,I)
G
contains an odd cycle
So we may assume that
or
G
{x,y,z}.
G.
#
C2n+l.
contains only triangles.
H4
By Lema 2.2, G # K4, hence
L2(a,b)
If
n
2
Let
C
be any
3
then
Consider two cases:
It follows that either
3.
L2(O,I
Again Lemma 2.2, G
G
G
and we are
K
G
4
for all non-negatlve integers
implies that one of the graphs
H 2, H 4, or H
6
implies
and
b.
is contained in
G.
a
This completes the proof of case (1).
(li)
G
Assume that at least one of the vertices x,y,z is of degree 2. By Lema 2.2,
L
or L2(0,b) or L3(a,b) for any non-negatlve integers
is not a subgraph of
a, b, and c; hence
G must contain one of the graphs
completes the proof of case (ll) and of the theorem.
H
H 2, H
3, or H 5.
This
REFERENCES
I.
2.
3.
BIALOSTOCKI, A. An Analog of the ErdBs-Szekeres Theorem. Submitted.
CALDERBANK, A.R., CHUNG, F.R.K. and STURTEVANT, D.G. Increasing Sequences with
Nonzero Block Sums and Increasing Paths in Edge-Ordered Graphs, Discrete
Math 50(1984), 15-28.
CHVTAL, V. and KOMLS, J. Some Combinatorial Theorems on Monotonlclty, Canad.
Math. Bull. 14(1971), 151-157.